Effective Mass

3 Key excerpts on "Effective Mass"

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  eBook - PDF

  Introductory Semiconductor Device Physics

  • Greg Parker(Author)
  • 2004(Publication Date)
  • CRC Press

    (Publisher)

  In some metals there may also be a few electrons in states above the hole states leading to bipolar conduction! These examples, although interesting, are now going a little further into band theory than I wish to cover. 2.5 The concept of Effective Mass It cannot come as a surprise that an electron moving through a crystal does not behave in the same way as an electron travelli l lg in a vacuum. If we apply an external force to an electron in a crystal , say by applying a voltage across the crystal, then the electron will accelerate , but at a different rate to a free electron experiencing the same electric field. This is not surprising; the electron in the crystal has to try and make its way through the crystal lattice, interacting with all the different potentials it would see depending on its direction through the lattice. What is surprising is that the complex interaction of the electron with the lattice can be taken into account, and we can use free-electron -type equations if we simply change the mass of the electron from its actual (vacuum) mass. This altered mass is called the electron's Effective Mass in the crystal and it depends on the direction it is travelling in the crystal. Although we don't have a free hole mass to compare with a hole in a crystal, the hole mass is also dependent on the direction in which the hole is travelling within the crystal. Why do we need to know about Effective Masses? There are at least four good reasons. 1 Up to this point in the chapter we have only considered energy-band diagrams. To appreciate what the Effective Mass means we have to look at energy-momentum diagrams. These energy-momentum diagrams will later show us why bulk silicon is a poor producer of photons and why gallium arsenide produces photons with a high efficiency.

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  eBook - ePub

  Solid State Physics

  • Siegfried Hunklinger, Christian Enss(Authors)
  • 2022(Publication Date)
  • De Gruyter

    (Publisher)

  9  Electronic Transport Properties

  The knowledge of the electronic density of states, the application of Fermi statistics and the inclusion of the interaction of the electrons with the periodic lattice potential has allowed us to explain a number of fundamental static solid state properties. We now turn to phenomena in which the motion of the electrons plays a crucial role. As we will see, the periodic modulation of the lattice potential and the associated band structure has rather amazing consequences for electron transport. Two of these properties have already been mentioned in the previous chapter without further explanation: fully occupied bands make no contribution to the electrical conductivity and the Effective Mass of the electrons can be positive or negative. In this chapter, we develop the concept of the Effective Mass (see also Section 8.4) and the concept of positively charged holes. Both of these concepts will be of great importance in later chapters. Subsequently, we will discuss charge and heat transport within metals. Particularly interesting phenomena are observed when the sample is located in a magnetic field. Since electrons in magnetic fields move on surfaces of constant energy, such experiments can be used to determine the shape of Fermi surfaces. Furthermore, magnetic fields restrict the motion of the electrons causing the quantization of their orbits. Consequences of this effect become especially clear in connection with the quantum Hall effect, which we discuss at the end of the chapter.

  9.1  Equation of Motion and Effective Mass

  9.1.1  Electrons as Wave Packets

  When deriving the density of states, we treated electrons as waves. The question arises: to what extent are classical equations (such as Newton’s second law) applicable when considering wave-like electrons in solids? It is known from quantum mechanics that particles cannot be precisely localized simultaneously in position and momentum space. Since the inequality δkδr > 1 must always be fulfilled, the position vector r and the wave vector k cannot be simultaneously and accurately defined. However, a limited localization of the wave function with respect to position and momentum can be achieved by the superposition of states with different wave vectors. For example, free electrons can be represented as wave packets by the superposition of plane waves. The wave function ψ(r

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  Fundamentals Of Semiconductor Physics And Devices

  • Rolf Enderlein, Norman J Horing(Authors)
  • 1997(Publication Date)
  • World Scientific

    (Publisher)

  Fields which change only little over this very small distance can be considered as homogeneous. In addition to excluding spatial inhomogeneities of the electric field, we will also exclude temporal changes in our discussion below. This only means that the fre- quencies of these changes should be small compared with the characteristic frequencies of the electrons of the semiconductor. 3.8.1 Effective Mass equation and stationary electron states A semiconductor in a homogeneous external electric field E represents a perturbed crystal in the sense of sections 3.1 and 3.2. The presence of such a field can be described by adding a perturbation potential V’(x) to the one-electron Hamiltonian H of the ideal crystal. This potential is defined as the difference of the energy of a crystal electron in the presence of the electric field and without it, and thus is given by V’(x) = eE . x. (3.294) As in the case of point perturbations considered in sections 3.4 and 3.5, the perturbation potential V’(x) of equation (3.294) does not possess lattice translation symmetry. Moreover, V’(x) diverges at infinity. The latter fact implies that, unlike to the case of point perturbations, an infinite crystal in a homogeneous electric field cannot be replaced by a perturbed supercell whose periodic repetition forms an unperturbed supercrystal. The crystal in an electric field has to be treated as what it in fact is, namely an infinite system with 2-dimensional lattice symmetry perpendicular to the field, and no lattice symmetry along the field direction. If the field strength E is not too large then the perturbation potential (3.295) fulfills the smoothness condition of Effective Mass theory of section 3.3, which here reads e I E I a << Eg. (3.295) With this condition fulfilled, the Effective Mass equation (3.53) is applicable. Here it has the form 3.8. Macroscopic electric fields 43 5 {Ey(-iV) +eE.x}FV~,(x) = EvFv~,(x).

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